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I am going to take a calculus course soon and everyone tells me that it is very difficult. What is considered difficult about it and why do so many people fail in calculus courses?I am asking these questions so that I can prepare myself beforehand and do not face any difficulties which most people face.

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    $\begingroup$ If, like myself, you find calculational gruntwork more "difficult" than conceptual complexity, you may find calculus much easier than precalc. $\endgroup$ Sep 14, 2016 at 18:47
  • $\begingroup$ I really don't think calculus itself or the underlying concepts are that difficult. The problem comes up when you start dealing with transcendental numbers (tan, sin, cos, etc.) in which case the algebra part of it gets hard, like it always does when dealing with those values. $\endgroup$ Feb 5, 2018 at 20:54

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Part of it is that a lot of adults now never took calculus, because calculus wasn't as commonly offered in high schools as it is now. In their era, it was a college level subject. So they see it as super high level.

People fail in calculus courses because it is at a slightly higher conceptual level than pre-calculus and (high school) algebra. Calculus requires that you put in a lot of work doing practice problems, which is something a lot of people aren't willing to do.

Ultimately though, calculus is a bogeyman of sorts. It's really not the devil it's made out to be. For someone who did well in pre-calculus, calculus would just be the next progression, and it wouldn't seem like some huge jump up in difficulty. The real abrupt change in mathematics is from computation to proofs. I'm assuming your class isn't proof-based.

So, to not fail, just read your textbook, pay attention in class, and do practice problems. Normal behavior. You'll be fine, really.

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    $\begingroup$ I disagree that it is a "slightly higher conceptual leap;" I think there is a reason much or precalculus has been around since Greece and babylon while in Calculus you will use things proved in the last 100-200 years. That being said, of course you can succeed by reading the textbook in most intro courses $\endgroup$ Sep 14, 2016 at 17:17
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    $\begingroup$ @qbert Disagree with what you say, I think the reasons are purely historical, Archimedes was in possession of many calculus type concepts. The roman empire was unconcerned with pure science, and, well I dont need to repeat here the history of europe. $\endgroup$ Sep 14, 2016 at 17:27
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    $\begingroup$ @qbert Logarithms were discovered in the 17th century, and those are typically studied in precalculus. $\endgroup$ Sep 14, 2016 at 18:45
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In my opinion, Calculus is peppered with nonintuitive facts. These facts usually relate to infinity or thinking in the limit; calculus is often the first time you confront either of those two.

In support of my opinion is the fact that calculus was a massive human undertaking, spanning decades during which many super smart people "stood on each other's shoulders" and ignored complaints from the scientific community. There were doubts taking limits or using infinitesimals was valid and repeatedly, things people thought were true turned out not to be. Continuity implies differentiability? Nope: The Weirstrass function fails this, and badly. Think there can only be 1 "size" of infinity? Cue the continuum.

A pair of examples that I still don't find intuitive, one which you will see this year, the other you will see later in calculus:

Why does $\sum_{n=1}^{\infty}\frac{1}{n}=\infty$ but $\sum_{n=1}^{\infty}\frac{1}{n^2}$ a finite number (not to mention why it is actually equal to $\frac{\pi^2}{6}$).

Why is $\sqrt{x}$ uniformly continuous on the interval $[0,1]$, while $1/x$ on $(0,1)$ is not? They both have unbounded slope for some time on intervals of the same length.

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  • $\begingroup$ Any function on a compact subset of R is uniformly continuous. Another way to see it is that for $\sqrt{x}$ on [0,1], there is a maximum amount that it can grow by at any value in [0,1] (its derivative is bounded). However, for 1/x, its derivative as $x\to 0$ is unbounded. In other words, as x gets closer to 0, the function 1/x grows at an increasing rate, so it's continuity isn't uniform. $\endgroup$
    – Jake
    Sep 14, 2016 at 17:21
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    $\begingroup$ First: Yes, I know any continuous function on a compact set is uniformly continuous, but appealing to a theorem from analysis is hardly providing intuition. Secondly, as I mentioned in my post, $\sqrt{x}$ does not have bounded derivatiave on $[0,1]$ as it grows infinitely quickly near zero. $\endgroup$ Sep 14, 2016 at 17:25
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    $\begingroup$ @Jake But my whole point is that it is non intuitive! Appealing l to measure theory for intuition for what is a fairly elementary fact from calculus further serves my point $\endgroup$ Sep 14, 2016 at 17:32
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    $\begingroup$ From an elementary calculus perspective, why are $(0,1)$ and $[0,1]$ so different? In fact people learning calculus get the two mixed up all the time $\endgroup$ Sep 14, 2016 at 17:33
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    $\begingroup$ In terms of topology, I think the easiest way to see a major difference between (0,1) and [0,1] is to draw functions with those intervals as the domain. I think within a few minutes any student would reasonably agree that the function whose domain is [0,1] is bounded and if a function on (0,1) is unbounded it's because the function approaches infinity at either 0 or 1, but nowhere else. With those two ideas in mind, I think it settles 1/x isn't uniformly continuous on (0,1). It all goes back to the rectangles from the wikipedia page. I do agree with you on the infinite sums though! $\endgroup$
    – Jake
    Sep 14, 2016 at 18:01
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One of the difficulties with calculus is the amount of time spent solving one problem. Most of that time is simplifying, evaluating, and the like, before applying the calculus-level concept at the end. If you can still do precalculus-level work fluently and without many mistakes, then the exercises won't take too long; it's when students get bogged down by loss of fluency that they get frustrated with how long it takes for one exercise (or exam question).

One other piece of advice that I haven't seen in the answers posted before this one: go to your instructor's office hours, even if you have no specific questions to ask. Just watch as others ask for help (and receive it). Doing that was a huge help for me (though I got a median-level score on the midterm, I got the highest score on the final), so I recommend it for all college math courses, but especially for calculus.

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When you first learn arithmetic, adding / subtracting / multiplying / division / exponentiation / roots ... that seems hard at first. With practice, it gets easier. Multiplication is just repeated addition. Exponentiation is just repeated multiplication. As such, you could say that multiplication is just "meta" addition (it takes addition as a starting point and builds on it), exponentiation is just "meta" multiplication, roots are just "meta" division.

Many people can do these based on rote (memorizing multiplication tables, etc.). They don't have to go looking for patterns. So they never do.

When you're studying algebra, the arithmetic is used but it's "background" compared to what you're learning. Instead of doing those things on numbers, you're doing the on numbers and variables. And you're finding meta-patterns which can be useful "shortcuts" for arithmetic

Pop quiz: what's $41 \cdot 39$? Gee, that's $(40 - 1)(40 + 1)$ which is $40^2 - 1$

In that regard, algebra can be seen as "meta" arithmetic.

If your mind isn't already looking for "meta," algebra is difficult and everything above it is even more so.

Trig / pre-calc introduces you to functions, including the usual sine / cosine/ tangent and logarithm. This function applied across this entire expression gives you this other expression ($sin(2\theta) = ?$). This expression using this function can be expressed in this other way, which may be easier to solve. Logarithms are just "meta" exponentiation / roots.

Calculus is meta algebra. You're not just manipulating numbers (arithmetic), variables (algebra) and functions (trig / pre-calc). You're manipulating entire formulas / expressions which use ALL of the above. Doing a differential (derivative) or an integral on an equation gives you a different equation, with useful properties relative to what you started with. The rules for getting from A to B ... that's really hard to do by rote. You need to be looking for patterns, learning those, learning how to apply those. You're not just going "meta," you're going "meta" on something which is already "meta." And that can be a big leap.

If you have an easier time with the lower-level stuff, your mind likely has already started looking for the meta-patterns in those lower levels. Calculus likely won't be such a stretch for you.

If your mind isn't already looking for meta-patterns, calculus can be really tedious. In my experience, the vast majority of people are NOT looking for meta-patterns, which can account for why so many people struggle with calculus and, indeed, any of the higher math.

This is further compounded by the fact that calculus tends to be taught as a stand-alone topic. If you aren't taking physics at the same time, you end up with one overriding question ... when am I EVER going to use / need this? And we tend to discount things we don't think we'll use / need. That's just human nature. Consequently, if you are taking calculus, try to see if you can get a physics class at the same time. I was lucky in that I was able to take both, concurrently, in High School.

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The problems are

1) Many students come out of high school with weak algebra and trig skills. Most calculus problems require good deal of algebra done quickly and correctly. So this is what you should work on.

2) If you have calculus in high school it is probably a watered down version that works against you. (Places vary, but in Germany is terrible.)

3) It depends on where you go to college but if you go to a "good" school you will have at 150-200+ students in your class, making the lectures a waste of time (unless, of course, you like chatting and texting).

4)Success in most lower level college and university courses is a matter of handing the assignments in on time and obeying all their little rules.

5)Use of technology, and computer labs have robbed many students of the small amount of personal contact with teachers that is left to them.

6)Your teachers DO NOT CARE. ( I know this because I spent a lot of time behind the scenes. And if they do claim to care they are saying it for political reasons. And if the really do care, they wont be coming back next semester.)

BUT never forget Calculus is beautiful, one of the most elegant and creative discoveries of all time. Just full of wonderful formulas. Too bad you are now to be surrounded by people who hate it.

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  • $\begingroup$ 1) is very true. If you're using all your time just keeping up with the algebra, you won't be able to pay attention to the calculus. $\endgroup$
    – John Feltz
    Sep 14, 2016 at 17:38
  • $\begingroup$ @JohnFeltz Yeah it really the major problem for most students. $\endgroup$ Sep 14, 2016 at 17:44
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    $\begingroup$ Disagree strongly with #6. I never had a college professor that didn't take the time to help me with what I was missing. And I have 3 degrees. $\endgroup$
    – scott
    Sep 14, 2016 at 18:26
  • $\begingroup$ @scott Hi Scott: And are they still working there ? To be serious, the fact that you have 3 degrees says that you are willing to sign up for more. In modern terminology you are a repeat customer. Of course they will be nice to you, they are grooming you. Also in modern terminology: respect. $\endgroup$ Sep 14, 2016 at 18:37
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    $\begingroup$ I've never really encountered a math teacher or professor that did not care. As long as you turn in homework/assignments when due, are attentive in class (e.g. making eye contact with instructor, taking notes), and are willing to visit the instructor during office hours if you are struggling, or simply stuck on one problem, or want clarification...etc. $\endgroup$
    – amWhy
    Sep 14, 2016 at 18:42

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